In this paper, we consider the surface area preserving mean curvature flow in quasi-Fuchsian 3-manifolds. We show that the flow exists for all times and converges exponentially to a smooth surface of constant mean curvature with the same surface area as the initial surface.
In this paper, we investigate a horizontal Laplacian version of the clamped plate problem on Carnot groups and obtain some universal inequalities. Furthermore, for the lower order eigenvalues of this eigenvalue problem on carnot groups, we also give some universal inequalities.
In this paper, the relationship between the existence of closed geodesics and the volume growth of complete noncompact Riemannian manifolds is studied. First the authors prove a diffeomorphic result of such an n-m2nifold with nonnegative sectional curvature, which improves Marenich-Toponogov＇s theorem. As an application, a rigidity theorem is obtained for nonnegatively curved open manifold which contains a clesed geodesic. Next the authors prove a theorem about the nonexistence of closed geodesics for Riemannian manifolds with sectional curvature bounded from below by a negative constant.
